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This question paper consists of 6 printed pages and 2 blank pages.

OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4762Mechanics 2

Friday 27 JANUARY 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.

• Answer all the questions.

• You are permitted to use a graphical calculator in this paper.

• Final answers should be given to a degree of accuracy appropriate to the context.

• The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when anumerical value is needed, use g = 9.8.

INFORMATION FOR CANDIDATES

• The number of marks is given in brackets [ ] at the end of each question or part question.

• You are advised that an answer may receive no marks unless you show sufficient detail of theworking to indicate that a correct method is being used.

• The total number of marks for this paper is 72.

HN/3© OCR 2006 [A/102/2654] Registered Charity 1066969 [Turn over

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2

1 When a stationary firework P of mass 0.4 kg is set off, the explosion gives it an instantaneousimpulse of 16 N s vertically upwards.

(i) Calculate the speed of projection of P. [2]

While travelling vertically upwards at 32ms –1, P collides directly with another firework Q, of mass0.6 kg, that is moving directly downwards with speed um s –1, as shown in Fig. 1. The coefficientof restitution in the collision is 0.1 and Q has a speed of 4 m s –1 vertically upwards immediatelyafter the collision.

Fig. 1

(ii) Show that and calculate the speed and direction of motion of P immediately after thecollision. [7]

Another firework of mass 0.5kg has a velocity of , where i and j are horizontal

and vertical unit vectors, respectively. This firework explodes into two parts, C and D. Part C has

mass 0.2 kg and velocity immediately after the explosion.

(iii) Calculate the velocity of D immediately after the explosion in the form . Show that Cand D move off at 90° to one another. [8]

aib j

(3i4 j) m s1

(3.6i 5.2 j) m s1

u 18

P0.4kg

ums-1

32ms-1

Q0.6kg

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3

2 A uniform beam, AB, is 6m long and has a weight of 240N.

Initially, the beam is in equilibrium on two supports at C and D, as shown in Fig. 2.1. The beam ishorizontal.

Fig. 2.1

(i) Calculate the forces acting on the beam from the supports at C and D. [4]

A workman tries to move the beam by applying a force T N at A at 40° to the beam, as shown inFig. 2.2. The beam remains in horizontal equilibrium but the reaction of support C on the beam iszero.

Fig. 2.2

(ii) ( A) Calculate the value of T . [4]

( B) Explain why the support at D cannot be smooth. [1]

The beam is now supported by a light rope attached to the beam at A, with B on rough, horizontalground. The rope is at 90° to the beam and the beam is at 30° to the horizontal, as shown in Fig. 2.3.The tension in the rope is PN. The beam is in equilibrium on the point of sliding.

Fig. 2.3

(iii) ( A) Show that and hence, or otherwise, find the frictional force between the beamand the ground. [5]

( B) Calculate the coefficient of friction between the beam and the ground. [5]

P = 60 3

B

P N

30∞

A

A

C D

B

T N

40∞

1m 2m6m

AC D

B

1m 2m6m

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4

3 (a) A uniform lamina made from rectangular parts is shown in Fig. 3.1. All the dimensions arecentimetres. All coordinates are referred to the axes shown in Fig. 3.1.

Fig. 3.1

(i) Show that the x-coordinate of the centre of mass of the lamina is 6.5 and find the y-coordinate. [5]

A square of side 2 cm is to be cut from the lamina. The sides of the square are to be parallelto the coordinate axes and the centre of the square is to be chosen so that the x-coordinate of

the centre of mass of the new shape is 6.4.

(ii) Calculate the x-coordinate of the centre of the square to be removed. [3]

The y-coordinate of the centre of the square to be removed is now chosen so that the y-coordinate of the centre of mass of the final shape is as large as possible.

(iii) Calculate the y-coordinate of the centre of mass of the lamina with the square removed,giving your answer correct to three significant figures. [3]

2

10

6

14

2

10

4

O

y

x

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5

(b) Fig. 3.2 shows a framework made from light rods of length 2m freely pin-jointed at A, B, C,D and E. The framework is in a vertical plane and is supported at A and C. There are loads of 120 N at B and at E. The force on the framework due to the support at A is RN.

each rod is2m long

Fig. 3.2

(i) Show that [2]

(ii) Draw a diagram showing all the forces acting at the points A, B, D and E, including theforces internal to the rods.

Calculate the internal forces in rods AE and EB, and determine whether each is a tensionor a thrust. [You may leave your answers in surd form.] [6]

(iii) Without any further calculation of the forces in the rods, explain briefly how you can tellthat rod ED is in thrust. [1]

[Question 4 is printed overleaf.]

R 150.

AB

C

E D

R N

120N 120N

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4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During6 seconds, the work done against resistances is 510 J and the speed of the block increases from

to .

(i) Calculate the power of the pulling force. [4]

The block is now put on a rough plane that is at an angle to the horizontal, where The

frictional resistance to sliding is 11gN. Alight string parallel to the plane is connected to the block.

The string passes over a smooth pulley and is connected to a freely hanging sphere of mass m kg,

as shown in Fig. 4.

Fig. 4

In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speedof vertically downwards. The block never reaches the pulley.

(ii) Suppose that and that after the sphere is released the block moves x m up the plane

before coming to rest.

( A) Find an expression in terms of x for the change in gravitational potential energy of thesystem, stating whether this is a gain or a loss. [4]

( B) Find an expression in terms of x for the work done against friction. [1]

(C ) Making use of your answers to parts ( A) and ( B), find the value of x. [3]

(iii) Suppose instead that . Calculate the speed of the sphere when it has fallen a distance0.5 m from its point of release. [4]

m 15

m 5

4 m s1

2 0 k g

s t r i n g

mkg

a

m o t i o n

11g N

sin a 35 .a

8 m s15 m s1

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This question paper consists of 6 printed pages and 2 blank pages.

OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4762Mechanics 2

Monday 19 JUNE 2006 Morning 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes


• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.




• The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when anumerical value is needed, use g = 9.8.





HN/4© OCR 2006 [A/102/2654] Registered Charity 1066969 [Turn over

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4762 June 2006 [Turn over

3

1 (a) Two small spheres, P of mass 2 kg and Q of mass 6 kg, are moving in the same straight linealong a smooth, horizontal plane with the velocities shown in Fig. 1.1.

Fig. 1.1

Consider the direct collision of P and Q in the following two cases.

(i) The spheres coalesce on collision.

( A) Calculate the common velocity of the spheres after the collision. [3]

( B) Calculate the energy lost in the collision. [2]

(ii) The spheres rebound with a coefficient of restitution of in the collision.

( A) Calculate the velocities of P and Q after the collision. [6]

( B) Calculate the impulse on P in the collision. [2]

(b) A small ball bounces off a smooth, horizontal plane. The ball hits the plane with a speed of

at an angle of to it. The ball rebounds at an angle of to the plane, as

shown in Fig. 1.2.

Fig. 1.2

Calculate the speed with which the ball rebounds from the plane.

Calculate also the coefficient of restitution in the impact. [6]

arcsin 1213

arcsin 35

arcsin35arcsin

121326 m s

–1

23

Q6kg

2ms –1

P2kg

4ms –1

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4

2 Two heavy rods AB and BC are freely jointed together at B and to a wall at A. AB has weight 90 Nand centre of mass at P; BC has weight 75 N and centre of mass at Q. The lengths of the rods andthe positions of P and Q are shown in Fig. 2.1, with the lengths in metres.

Initially, AB and BC are horizontal. There is a support at R, as shown in Fig. 2.1. The system is

held in equilibrium by a vertical force acting at C.

Fig. 2.1

(i) Draw diagrams showing all the forces acting on rod AB and on rod BC.

Calculate the force exerted on AB by the hinge at B and hence the force required at C. [6]

The rods are now set up as shown in Fig. 2.2. AB and BC are each inclined at 60° to the verticaland C rests on a rough horizontal table. Fig. 2.3 shows all the forces acting on AB, including theforces X N and Y N due to the hinge at A and the forces U N and V N in the hinge at B. The rodsare in equilibrium.

(ii) By considering the equilibrium of rod AB, show that [3]

(iii) Draw a diagram showing all the forces acting on rod BC. [1]

(iv) Find a further equation connecting U and V and hence find their values. Find also the frictionalforce at C. [8]

60 3 3= +U V .

A

B

Q

C

P60∞60∞

P

90N X N

Y N

V N

U N

60∞

B

A

Fig. 2.3Fig. 2.2

A P R

12 2

B

10.5

Q C

90N 75N

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5

3 (a) A car of mass 900 kg is travelling at a steady speed of up a hill inclined at

to the horizontal. The power required to do this is 20 kW.

Calculate the resistance to the motion of the car. [4]

(b) A small box of mass 11 kg is placed on a uniform rough slope inclined at arccos 12 ––13

to thehorizontal. The coefficient of friction between the box and the slope is m .

(i) Show that if the box stays at rest then [3]

For the remainder of this question, the box moves on a part of the slope where .

The box is projected up the slope from a point P with an initial speed of It travels adistance of 1.5 m along the slope before coming instantaneously to rest. During this motion,the work done against air resistance is 6 joules per metre.

(ii) Calculate the value of v. [5]

As the box slides back down the slope, it passes through its point of projection P and laterreaches its initial speed at a point Q. During this motion, once again the work done against airresistance is 6 joules per metre.

(iii) Calculate the distance PQ. [6]


v m s –1.

m 0.2

m 512 .

arcsin 0.116 m s –1

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4 Fig. 4.1 shows four uniform rods, OA, AB, BE and CD, rigidly fixed together to form a frame. Therods have weights proportional to their lengths and these lengths, in centimetres, are shown inFig. 4.1.

Fig. 4.1

(i) Calculate the coordinates of the centre of mass of the frame, referred to the axes shown inFig. 4.1. [5]

The bracket shown in Fig. 4.2 is made of uniform sheet metal with cross-section the frame shownin Fig. 4.1. The bracket is 40 cm wide and its weight is 60 N. It stands on a horizontal planecontaining O x and O z.

Fig. 4.2

(ii) Write down the coordinates of the centre of mass of the bracket, referred to the axes shown inFig. 4.2. [2]

A force P N acts vertically downwards at the point M, shown in Fig. 4.2. M is the mid-point of EF.The bracket is on the point of tipping.

(iii) Calculate the value of P. [4]

40

O

B

F

C

xA D

yE

M

z

10

10 20

30

O

B EC

y

xA D

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In another situation, a horizontal force Q N acts through M parallel to EB and in the direction fromE to B. The value of Q is increased from zero with the bracket in equilibrium at all times.

(iv) Draw a diagram showing the forces acting on the bracket when it is on the point of tipping.[1]

(v) If the limiting frictional force between the bracket and the plane is 30N, does the bracket slideor tip first as Q is increased? [5]

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• Write your name, centre number and candidate number in the spaces provided on the answer booklet.




• The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when a

numerical value is needed, use g = 9.8.




ADVICE TO CANDIDATES

• Read each question carefully and make sure you know what you have to do before starting your

answer.

• You are advised that an answer may receive no marks unless you show sufficient detail of the

working to indicate that a correct method is being used.

This document consists of 7 printed pages and 1 blank page.

HN/5 © OCR 2007 [A/102/2654] OCR is an exempt Charity [Turn over

ADVANCED GCE UNIT 4762/01MATHEMATICS (MEI)

Mechanics 2

WEDNESDAY 10 JANUARY 2007 AfternoonTime: 1 hour 30 minutes

Additional materials: Answer booklet (8 pages)Graph paper MEI Examination Formulae and Tables (MF2)

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1 A sledge and a child sitting on it have a combined mass of 29.5kg. The sledge slides on horizontalice with negligible resistance to its movement.

(i) While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travellinghorizontally at 10 m s –1. The coefficient of restitution in the collision is 0.8. After the impactthe speeds of the sledge and the ball are and respectively.

Calculate and and state the direction in which the ball is travelling after the impact. [7]

(ii) While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travellinghorizontally at 10 m s –1. The snowball sticks to the sledge.

( A) Calculate the velocity with which the combined sledge and snowball start to move. [3]

( B) The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. Whathappens to the velocity of the sledge? Give a reason for your answer. [2]

(iii) In another situation, the sledge is travelling over the ice at 2 m s –1 with 10.5 kg of snow on it(giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge,parallel to the ground and in the positive direction of the motion of the sledge. Immediatelyafter the snowball is thrown, the sledge has a speed of and the snowball and sledgeare separating at a speed of 10 m s –1.

Draw a diagram showing the velocities of the sledge and snowball before and after thesnowball is thrown.

Calculate V . [5]

V m s1

V 2

V 1

V 2

m s1V 1

m s1

2

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2

Fig. 2

Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD,BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.]

The rods AB, BC and DE are horizontal.

The rods are freely pin-jointed to each other at A, B, C, D and E.

The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD.The pin-joint at D rests on this plane.

The following external forces act on the framework: a vertical load of L N at C; the normal reaction

force R N of the plane on the framework at D; the horizontal and vertical forces X N and Y N,respectively, acting at A.

(i) Write down equations for the horizontal and vertical equilibrium of the framework. [3]

(ii) By considering moments, find the relationship between R and L. Hence show thatand [4]

(iii) Draw a diagram showing all the forces acting on the pin-joints, including the forces internalto the rods. [2]

(iv) Show that the internal force in the rod AD is zero. [2]

(v) Find the forces internal to AB, CE and BC in terms of L and state whether each is a tensionor a thrust (compression). [You may leave your answers in surd form.] [7]

(vi) Without calculating their values in terms of L, show that the forces internal to the rods BD andBE have equal magnitude but one is a tension and the other a thrust. [2]

Y 0.

X L= 3

Y N

X N

LN

RN

60∞

BA C

D E

fixed

3

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3 A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are incentimetres. All the faces are made of the same uniform, rigid and thin material. All coordinatesrefer to the axes shown in this figure.

Fig. 3.1

(i) The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an openbox without a base or a top. Write down the coordinates of the centre of mass of this open box.[1]

The base OABC is added to the vertical faces.

(ii) Write down the x- and y-coordinates of the centre of mass of the box now. Show that the z-coordinate is now 1.875. [5]

The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. Thelid is open so that it hangs in a vertical plane touching the face FGCB.

(iii) Show that the coordinates of the centre of mass of the box in this situation are[6]

[This question is continued on the facing page.]

(10, 2.4, 2.1).

z

y

x20

5

4

D

G F

E

A

BC

O

4

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5

The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OAhorizontal and the base inclined at 30° to the horizontal, as shown in Fig. 3.2.

Fig. 3.2

The weight of the box is 40 N. A force P N acts parallel to the plane and is applied to themid-point of FG at 90° to FG. This force tends to push the box down the plane. The box does notslip and is on the point of toppling about the edge AO.

(iv) Show that the clockwise moment of the weight of the box about the edge AO is about0.411 Nm. [4]

(v) Calculate the value of P. [2]


30∞

G

CO

AB

F

horizontal

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4 Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of frictionbetween it and the existing roof is 0.75. The roof is at 30° to the horizontal and the bottom of theroof is 6 m above horizontal ground, as shown in Fig. 4.

Fig. 4

(i) Calculate the limiting frictional force between a tile and the roof.

A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.)[5]

(ii) The tiles are raised 6 m from the ground, the only work done being against gravity. They arethen slid 4m up the roof and placed at the point A shown in Fig. 4.

( A) Show that each tile gains 156.8 J of gravitational potential energy. [3]

( B) Calculate the work done against friction per tile. [2]

(C ) What average power is required to raise 10 tiles per minute from the ground to A? [2]

(iii) A tile is kicked from A directly down the roof. When the tile is at B, x m from the edge of theroof, its speed is 4m s –1. It subsequently hits the ground travelling at 9 m s –1. In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90J.

Use an energy method to find x. [5]

6m

4m

30∞

A

not toscale

6

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• Write your name, centre number and candidate number in the spaces provided on the answer booklet.




• The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when a





ADVICE TO CANDIDATES

• Read each question carefully and make sure you know what you have to do before starting your

answer.

• You are advised that an answer may receive no marks unless you show sufficient detail of the

working to indicate that a correct method is being used.

This document consists of 7 printed pages and 1 blank page.

HN/5 © OCR 2007 [A/102/2654] OCR is an exempt Charity [Turn over

ADVANCED GCE UNIT 4762/01MATHEMATICS (MEI)

Mechanics 2

WEDNESDAY 20 JUNE 2007 AfternoonTime: 1 hour 30 minutes

Additional materials: Answer booklet (8 pages)Graph paper MEI Examination Formulae and Tables (MF2)

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1 (a) Disc A of mass 6 kg is on a smooth horizontal plane. The disc is at rest and then a constantforce of magnitude 9N acts on it for 2 seconds.

(i) Find the magnitude of the impulse of the force on the disc. Hence, or otherwise, find thespeed of the disc after the two seconds. [2]

Without losing speed, disc A now collides directly with disc B of mass 2 kg which is also onthe plane. Just before the collision, disc B is travelling at 1 m s –1 in the opposite direction to themotion of A, as shown in Fig. 1.1. On impact the two discs stick together to form the combinedobject AB.

Fig. 1.1

(ii) Show that AB moves off with a speed of 2 ms –1 in the original direction of motion of disc A.[3]

(iii) Find the impulse that acts on disc B in the collision. [2]

The combined object AB now collides directly with disc C of mass 10 kg, which is moving onthe plane at 1.8m s –1 in the same direction as AB. After this collision the speed of AB is vm s –1

in the same direction as its speed before the impact, and disc C moves off with speed 1.9m s –1.

(iv) ( A) Draw a diagram indicating the velocities before and after the collision. [1]

( B) Calculate the value of v. [3]

(C ) Calculate the coefficient of restitution in the collision. [3]

(b) A small ball is thrown horizontally with a speed of . The point of projection is 10 mabove a smooth horizontal plane, as shown in Fig. 1.2.

Fig. 1.2

The coefficient of restitution in the impact between the ball and the plane is

Calculate the vertical component of the velocity of the ball immediately after its first impact

with the plane and also the angle at which the ball rebounds from the plane. [5]

47 .

10m

8 m s-1

8 1ms-

A6kg

B2kg

m1 s-1

3

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2 The position of the centre of mass, G, of a uniform wire bent into the shape of an arc of a circle of radius r and centre C is shown in Fig. 2.1.

Not toscale

Fig. 2.1

(i) Use this information to show that the centre of mass, G, of the uniform wire bent into the

shape of a semi-circular arc of radius 8 shown in Fig. 2.2 has coordinates . [3]

Not toscale

Fig. 2.2

A walking-stick is modelled as a uniform rigid wire. The walking-stick and coordinate axes areshown in Fig. 2.3. The section from O to A is a semi-circular arc and the section OB lies along the x-axis. The lengths are in centimetres.

Fig. 2.3

(ii) Show that the coordinates of the centre of mass of the walking-stick are correct

to two decimal places. [6]

(25.37, 2.07),

y

x

16

72

Not toscale

O

A

B

G

8

16

y

xO

-Ê Ë ˆ

¯ 16

8p

,

q q

G

C

r CG is

rsinq

q

where q is in radians

4

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5

The walking-stick is now hung from a shelf as shown in Fig. 2.4. The only contact between thewalking-stick and the shelf is at A.

Fig. 2.4

(iii) When the walking-stick is in equilibrium, OB is at an angle a to the vertical.

Draw a diagram showing the position of the centre of mass of the walking-stick in relation to A.

Calculate a . [5]

(iv) The walking-stick is now held in equilibrium, with OB vertical and Astill resting on the shelf,by means of a vertical force, F N, at B. The weight of the walking-stick is 12N. Calculate F .

[3]

Not toscale

O

A

B

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© OCR 2007 4762/01 June 07

3 A uniform plank is 2.8m long and has weight 200 N. The centre of mass is G.

(i) Fig. 3.1 shows the plank horizontal and in equilibrium, resting on supports at A and B.

Fig. 3.1

Calculate the reactions of the supports on the plank at A and at B. [4]

(ii) Fig. 3.2 shows the plank horizontal and in equilibrium between a support at C and a peg at D.

Fig. 3.2

Calculate the reactions of the support and the peg on the plank at C and at D, showing thedirections of these forces on a diagram. [5]

Fig. 3.3 shows the plank in equilibrium between a support at P and a peg at Q. The plank is inclinedat to the horizontal, where and .

Fig. 3.3

(iii) Calculate the normal reactions at P and at Q. [6]

(iv) Just one of the contacts is rough. Determine which one it is if the value of the coefficient of friction is as small as possible. Find this value of the coefficient of friction. [4]

PG

0 .8 m

200N

Q

a

0 .2 m

0 .4 m

1.4 m

cos a 0.96sin a 0.28a

C G0.8m

0.4m

0.2m200N

D

C G0.8m

0.4m

0.2m200N

D 1.4m

B AG

0.8m 0.6m 0.2m 1.2m

200N

6

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© OCR 2007 4762/01 June 07

4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and waterhave total mass 20kg.

In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.

(i) How much work is done to raise the pail from rest so that it is travelling upwards atwhen at a distance of 4 m above its starting position? [4]

(ii) What power is required to raise the pail at a steady speed of [3]

Jack falls over and hurts himself. He then slides down a hill.

His mass is 35 kg and his speed increases from to while descending through avertical height of 3 m.

(iii) How much work is done against friction? [5]

In Jack’s further motion, he slides down a slope at an angle to the horizontal whereThe frictional force on him is now constant at 150 N. For this part of the motion, Jack’s initial speedis .

(iv) How much further does he slide before coming to rest? [5]

3 m s1

sin a 0.1.a

3 m s11 m s1

0.5 m s1?

0.5 m s1

7

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ADVANCED GCE 4762/01MATHEMATICS (MEI)

Mechanics 2

THURSDAY 17 JANUARY 2008 Afternoon

Time: 1 hour 30 minutes

Additional materials: Answer Booklet (8 pages)

Graph paper

MEI Examination Formulae and Tables (MF2)


• Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

• Read each question carefully and make sure you know what you have to do before startingyour answer.




• The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when a






This document consists of 6 printed pages and 2 blank pages.

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2

1 (a) A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along

horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-

ram has a mass of 4000 kg.

wall

Fig. 1.1

Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about

to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the

direction of its motion is 1500 N.

(i) At what speed does the battering-ram hit the wall? [3]

The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is

conserved and the coefficient of restitution in the impact is 0.2.

(ii) Calculate the speeds of the stone block and of the battering-ram immediately after the

impact. [6]

(iii) Calculate the energy lost in the impact. [3]

(b) Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed

1 8 m s−1 in the i direction. B has mass 8 kg and speed 9 m s−1 in the direction shown in Fig. 1.2,where i and j are the standard unit vectors.

A

4 kg

B8 kg

18 m s –1

9 m s –1

60°

Fig. 1.2

i

j

(i) Write down the linear momentum of A and show that the linear momentum of B is

(36i + 36√

3 j) N s. [2]

After the objects meet they stick together (coalesce) and move with a common velocity of

(ui + v j)m s−1.

(ii) Calculate u and v. [3](iii) Find the angle between the direction of motion of the combined object and the i direction.

Make your method clear. [2]

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3

2 A cyclist and her bicycle have a combined mass of 80 kg.

(i) Initially, the cyclist accelerates from rest to 3 m s−1 against negligible resistances along ahorizontal road.

( A) How much energy is gained by the cyclist and bicycle? [2]

( B) The cyclist travels 12 m during this acceleration. What is the average driving force on the

bicycle? [2]

(ii) While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from

4 m s−1 to 10 m s−1. During this motion, the total work done against friction is 1600 J and thedrop in vertical height is h m.

Without assuming that the hill is uniform in either its angle or roughness, calculate h. [5]

(iii) The cyclist reaches another horizontal stretch of road and there is now a constant resistance to

motion of 40 N.

( A) When the power of the driving force on the bicycle is a constant 200 W, what constant speed

can the cyclist maintain? [3]

( B) Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s−1

with an acceleration of 2 m s−2. [5]

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4

3

A

BC

DZ

E

F G

HIOJ

100 100

20 20

20 20

50 50

y

z

J A

D

Z

E

IH

O x

y

z

Fig. 3.1 Fig. 3.2

A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and

FGHI are all rectangles. The lengths of the sides are shown in centimetres.

(i) Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1.

[5]

The rectangles BCJA and FGHI are folded through 90◦ about the lines CJ and FI respectively to givethe fire-screen shown in Fig. 3.2.

(ii) Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown inFig. 3.2, are (2.5, 0, 57.5). [4]

The x - and y-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force

PN is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive

x direction. The fire-screen is on the point of tipping about the line AH.

(iii) Calculate the value of P. [5]

The coefficient of friction between the fire-screen and the floor is µ .

(iv) For what values of µ does the fire-screen slide before it tips? [4]

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5

4 Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m. The beam is freely pivoted

on a fixed support at D and is supported at C. The distance CD is 2.75 m.

C C

D D

E E

2.75 m 2.75 m1.75 m 1.75 m

1.5 m

R

440 N W N

Fig. 4.1 Fig. 4.2

P Q

The beam is horizontal and in equilibrium.

(i) Show that the anticlockwise moment of the weight of the beam about D is 1100 N m.

Find the value of the normal reaction on the beam of the support at C. [6]

The support at C is removed and spheres at P and Q are suspended from the beam by light strings

attached to the points C and R. The sphere at P has weight 440 N and the sphere at Q has weight W N.

The point R of the beam is 1.5 m from D. This situation is shown in Fig. 4.2.

(ii) The beam is horizontal and in equilibrium. Show that W = 1540. [3]

The sphere at P is changed for a lighter one with weight 400 N. The sphere at Q is unchanged. The

beam is now held in equilibrium at an angle of 20 ◦ to the horizontal by means of a light rope attachedto the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3.

C

D

E

2 .7 5 m

1 .7 5 m

1 .5 m

R 400 N

1540 N

Fig. 4.3

20°P

Q

(iii) Calculate the tension in the rope when it is

( A) at 90◦ to the beam, [6]

( B) horizontal. [3]

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ADVANCED GCE 4762/01MATHEMATICS (MEI)

Mechanics 2

MONDAY 16 JUNE 2008 Afternoon

Time: 1 hour 30 minutes

Additional materials (enclosed): None

Additional materials (required):

Answer Booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)


• Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

• Read each question carefully and make sure you know what you have to do before startingyour answer.




• The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when anumerical value is needed, use g = 9.8.





This document consists of 6 printed pages and 2 blank pages.

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2

1 (a) Disc A of mass 6 kg and disc B of mass 0.5 kg are moving in the same straight line. The relative

positions of the discs and the i direction are shown in Fig. 1.1.

A

6 kg

B0.5 kg

i

Fig. 1.1

The discs collide directly. The impulse on A in the collision is −12i N s and after the collision Ahas velocity 3i m s−1 and B has velocity 11i m s−1.

(i) Show that the velocity of A just before the collision is 5i m s−1 and find the velocity of B at

this time. [5]

(ii) Calculate the coefficient of restitution in the collision. [3]

(iii) After the collision, a force of −2i N acts on B for 7 seconds. Find the velocity of B after thistime. [4]

(b) A ball bounces off a smooth plane. The angles its path makes with the plane before and after the

impact are α and β , as shown in Fig. 1.2.

i

j

Fig. 1.2

The velocity of the ball before the impact is ui − v j and the coefficient of restitution in the impactis e.

Write down an expression in terms of u, v, e, i and j for the velocity of the ball immediately after

the impact. Hence show that tan β = e tanα . [5]

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3

2

x

y

O OA A

B BC C

D D2 2

6 6

6 6

6 6

Fig. 2.1 Fig. 2.2

Q P

A uniform wire is bent to form a bracket OABCD. The sections OA, AB and BC lie on three sides of a square and CD is parallel to AB. This is shown in Fig. 2.1 where the dimensions, in centimetres, are

also given.

(i) Show that, referred to the axes shown in Fig. 2.1, the x -coordinate of the centre of mass of the

bracket is 3.6. Find also the y-coordinate of its centre of mass. [6]

(ii) The bracket is now freely suspended from D and hangs in equilibrium.

Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the

vertical. [5]

(iii) The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D,as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal.

Calculate the tensions in the strings BP and DQ. [4]

The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a

straight line. This section is made of the same type of wire and has length L cm, as shown in Fig. 2.3.

x

y

O A

BC

D2

6

6

6

Fig. 2.3

E

L

not toscale

The value of L is chosen so that the centre of mass is now on the y-axis.

(iv) Calculate L. [4]

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4

3 (a) Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods AB, BC, AD and

BD. The rods are freely pin-jointed to each other at A, B and D and to a vertical wall at C and

D. There are vertical loads of L N at A and 3 LN at B. Angle DAB is 30◦, angle DBC is 60◦ andABC is a straight, horizontal line.

A B C

D

L N 3 N L

30° 60°

Fig. 3.1

(i) Draw a diagram showing the loads and the internal forces in the four rods. [2]

(ii) Find the internal forces in the rods in terms of L, stating whether each rod is in tension or in

thrust (compression). [You may leave answers in surd form. Note that you are not required

to find the external forces acting at C and at D.] [9]

(b) Fig. 3.2 shows uniform beams PQ and QR, each of length 2lm and of weightW N. The beams are

freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at 60 ◦ to

the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal.This force, U N, is shown in Fig. 3.3.

P R

Q Q

60° 60°

Fig. 3.2 Fig. 3.3

U N

Show that the frictional force between the floor and each beam is

√ 3

6W N. [7]

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5

4 (a)

A

B

0.15 kg

2 m

Fig. 4

40°

A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing

line of length 2 m. The other end of the line, A, is fixed and the line can swing freely.

The sphere swings with the line taut from a point where the line is at an angle of 40 ◦ with thevertical, as shown in Fig. 4.

(i) Explain why no work is done on the sphere by the tension in the line. [1]

(ii) Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the

lowest point of its swing and calculate the amount of gravitational potential energy lost when

it is at this point. [4](iii) Assuming that there is no air resistance and that the sphere swings from rest from the position

shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing. [2]

(iv) Now consider the case where

• there is a force opposing the motion that results in an energy loss of 0.6 J for everymetre travelled by the sphere,

• the sphere is given an initial speed of 2.5 m s−1 (and it is descending) with AB at 40◦ tothe vertical.

Calculate the speed of the sphere at the lowest point of its swing. [5]

(b) A block of mass 3 kg slides down a uniform, rough slope that is at an angle of 30◦ to the horizontal.

The acceleration of the block is 18g.

Show that the coefficient of friction between the block and the slope is 14

√ 3. [6]

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ADVANCED GCE

MATHEMATICS (MEI) 4762Mechanics 2

Candidates answer on the Answer Booklet

OCR Supplied Materials:

• 8 page Answer Booklet• Graph paper• MEIExamination Formulae and Tables (MF2)

Other Materials Required:

None

Friday 9 January 2009

Morning

Duration: 1 hour 30 minutes

**44776622**


• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.

• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions.• Do not write in the bar codes.• You are permitted to use a graphical calculator in this paper.• Final answers should be given to a degree of accuracy appropriate to the context.

• The acceleration due to gravity is denoted by g m s−2

. Unless otherwise instructed, when a numerical value isneeded, use g = 9.8.


• The number of marks is given in brackets [ ] at the end of each question or part question.• You are advised that an answer may receive no marks unless you show sufficient detail of the working to

indicate that a correct method is being used.• The total number of marks for this paper is 72.• This document consists of 8 pages. Any blank pages are indicated.

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2

1 (i) What constant force is required to accelerate a particle of mass m kg from rest to 2um s−1 in

5 seconds? [3]

Two discs P and Q are moving in the same straight line over a smooth, horizontal surface. Fig. 1 shows

the masses (in kg) and the velocities (in m s−1) of the discs before and after they collide directly. The

collision is perfectly elastic.

P Q

m 3m

2u u

vQvP

before

after

barrier

Fig. 1

(ii) Show that vQ

= 3

2u and find the value of v

P. [6]

After the collision, Q hits the barrier shown in Fig. 1 which is perpendicular to its path and bounces

back directly. The coefficient of restitution in this collision is e. Q collides again with P and on this

occasion they stick together (coalesce) to form a single object, R, that has a speed of 14um s−1 away

from the barrier.

(iii) Write down an expression in terms of e and u for the velocity of Q after collision with the barrier.

Find, in either order, the value of e and the impulse on the barrier. [9]

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3

2 One way to load a box into a van is to push the box so that it slides up a ramp. Some removal men are

experimenting with the use of different ramps to load a box of mass 80 kg.

80 kg

ground

rampvan floor

Fig. 2

Fig. 2 shows the general situation. The ramps are all uniformly rough with coefficient of friction 0.4

between the ramp and the box. The men push parallel to the ramp. As the box moves from one end

of the ramp to the other it travels a vertical distance of 1.25 m.

(i) Find the limiting frictional force between the ramp and the box in terms of θ . [3]

(ii) From rest at the bottom, the box is pushed up the ramp and left at rest at the top. Show that the

work done against friction is 392

tanθ J. [3]

(iii) Calculate the gain in the gravitational potential energy of the box when it is raised from the

ground to the floor of the van. [2]

For the rest of the question take θ =

35

◦

.

(iv) Calculate the power required to slide the box up the ramp at a steady speed of 1.5 m s−1. [4]

(v) The box is given an initial speed of 0.5 m s−1 at the top of the ramp and then slides down without

anyone pushing it. Determine whether it reaches a speed of 3 m s−1 while it is on the ramp. [5]

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4

3 A fish slice consists of a blade and a handle as shown in Fig. 3.1. The rectangular blade ABCD is of

mass 250 g and modelled as a lamina; this is 24 cm by 8 cm and is shown in the O xy plane. The handle

EF is of mass 125 g and is modelled as a thin rod; this is 30 cm long and E is attached to the mid-point

of CD. EF is at right angles to CD and inclined at α to the plane containing ABCD, where sin α = 0.6

(and cos α = 0.8). Coordinates refer to the axes shown in Fig. 3.1. Lengths are in centimetres. The y-

and -coordinates of the centre of mass of the fish slice are y and respectively.

O

A

D

C

B x

y

z

12

128

30

F

blade handle

Fig. 3.1

E

(i) Show that y = 9 13

and = 3. [8]

(ii) Suppose that the plane O xy in Fig. 3.1 is horizontal and represents a table top and that the fish

slice is placed on it as shown. Determine whether the fish slice topples. [2]

The ‘superior’ version of the fish slice has an extra mass of 125 g uniformly distributed over the

existing handle for 10 cm from F towards E, as shown in Fig. 3.2. This section of the handle may stillbe modelled as a thin rod.

O

A

D

C

B x

y

z F

Fig. 3.2

E

10

(iii) In this new situation show that y = 14 and = 6. [4]

A sales feature of the ‘superior’ version is the ability to suspend it using a very small hole in the blade.

This situation is modelled as the fish slice hanging in equilibrium when suspended freely about an

axis through O.

(iv) Indicate the position of the centre of mass on a diagram and calculate the angle of the line OE

with the vertical. [4]

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5

4 (a) A uniform, rigid beam, AB, has a weight of 600 N. It is horizontal and in equilibrium resting on

two small smooth pegs at P and Q. Fig. 4.1 shows the positions of the pegs; lengths are in metres.

A P Q B

1 12 2

600 N

Fig. 4.1

(i) Calculate the forces exerted by the pegs on the beam. [4]

A force of LN is applied vertically downwards at B. The beam is in equilibrium but is now on the

point of tipping.

(ii) Calculate the value of L. [3]

(b) Fig. 4.2 shows a framework in a vertical plane constructed of light, rigid rods AB, BC and CA.

The rods are freely pin-jointed to each other at A, B and C and to a fixed point at A. The pin-joint

at C rests on a smooth, horizontal support. The dimensions of the framework are shown in metres.

There is a force of 340 N acting at B in the plane of the framework. This force and the rod BC

are both inclined to the vertical at an angle α , which is defined in triangle BCX. The force on the

framework exerted by the support at C is R N.

A X B

C

17

8 8

15

17

R N

340 N

Fig. 4.2

(i) Show that R = 600. [4]

(ii) Draw a diagram showing all the forces acting on the framework and also the internal forces

in the rods. [2]

(iii) Calculate the internal forces in the three rods, indicating whether each rod is in tension or

in compression (thrust). [Your working in this part should correspond to your diagram in

part (ii).] [6]

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ADVANCED GCE



OCR Supplied Materials:

• 8 page Answer Booklet• Graph paper• MEIExamination Formulae and Tables (MF2)

Other Materials Required:

None

Thursday 11 June 2009

Morning


**44776622**





. Unless otherwise instructed, when a numerical value isneeded, use g = 9.8.





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2

1 (a) Two small objects, P of mass m kg and Q of mass km kg, slide on a smooth horizontal plane.

Initially, P and Q are moving in the same straight line towards one another, each with speed

um s−1.

After a direct collision with P, the direction of motion of Q is reversed and it now has a speed of 1

3um s

−1. The velocity of P is now vm s−1, where the positive direction is the original direction

of motion of P.

(i) Draw a diagram showing the velocities of P and Q before and after the impact. [1]

(ii) By considering the linear momentum of the objects before and after the collision, show that

v = 1 − 43k u. [3]

(iii) Hence find the condition on k for the direction of motion of P to be reversed. [2]

The coefficient of restitution in the collision is 0.5.

(iv) Show that v = −23u and calculate the value of k . [5]

(b) Particle A has a mass of 5 kg and velocity 32 m s−1. Particle B has mass 3 kg and is initially

at rest. A force 1−2

N acts for 9 seconds on B and subsequently (in the absence of the force),A and B collide and stick together to form an object C that moves off with a velocity V m s−1.

(i) Show that V = 3−1

. [4]

The object C now collides with a smooth barrier which lies in the direction 01. The coefficient

of restitution in the collision is 0.5.

(ii) Calculate the velocity of C after the impact. [3]

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3

2 (a) A small block of mass 25 kg is on a long, horizontal table. Each side of the block is connected to

a small sphere by means of a light inextensible string passing over a smooth pulley. Fig. 2 shows

this situation. Sphere A has mass 5 kg and sphere B has mass 20 kg. Each of the spheres hangs

freely.

25 kg

A 5 kg

20 kg B

Fig. 2

Initially the block moves on a smooth part of the table. With the block at a point O, the system

is released from rest with both strings taut.

(i) ( A) Is mechanical energy conserved in the subsequent motion? Give a brief reason for your

answer. [1]

( B) Why is no work done by the block against the reaction of the table on it? [1]

The block reaches a speed of 1.5 m s−1 at point P.

(ii) Use an energy method to calculate the distance OP. [5]

The block continues moving beyond P, at which point the table becomes rough. After travelling

two metres beyond P, the block passes through point Q. The block does 180 J of work against

resistances to its motion from P to Q.

(iii) Use an energy method to calculate the speed of the block at Q. [5]

(b) A tree trunk of mass 450 kg is being pulled up a slope inclined at 20◦ to the horizontal.

Calculate the power required to pull the trunk at a steady speed of 2.5 m s−1 against a frictional

force of 2000 N. [5]

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4

3 A non-uniform beam AB has weight 85 N. The length of the beam is 5 m and its centre of mass is 3 m

from A. In this question all the forces act in the same vertical plane.

Fig. 3.1 shows the beam in horizontal equilibrium, supported at its ends.

A B

85 N

3 m 2 m

Fig. 3.1

(i) Calculate the reactions of the supports on the beam. [4]

Using a smooth hinge, the beam is now attached at A to a vertical wall. The beam is held in equilibrium

at an angle α to the horizontal by means of a horizontal force of magnitude 27.2 N acting at B. This

situation is shown in Fig. 3.2.

3 m2.5 m

2 m 2 m85 N 85 N

27.2 N

A A

B B

Fig. 3.2 Fig. 3.3

34 N

0.5 m

(ii) Show that tan α = 15

8 . [4]

The hinge and 27.2 N force are removed. The beam now rests with B on a rough horizontal floor and

A on a smooth vertical wall, as shown in Fig. 3.3. It is at the same angle α to the horizontal. There is

now a force of 34 N acting at right angles to the beam at its centre in the direction shown. The beam

is in equilibrium and on the point of slipping.

(iii) Draw a diagram showing the forces acting on the beam.

Show that the frictional force acting on the beam is 7.4 N.

Calculate the value of the coefficient of friction between the beam and the floor. [10]

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5

4 In this question you may use the following facts: as illustrated in Fig. 4.1, the centre of mass, G, of a

uniform thin open hemispherical shell is at the mid-point of OA on its axis of symmetry; the surface

area of this shell is 2π r 2, where r is the distance OA.

O

G

AOG = GA

Fig. 4.1

A perspective view and a cross-section of a dog bowl are shown in Fig. 4.2. The bowl is made

throughout from thin uniform material. An open hemispherical shell of radius 8 cm is fitted inside an

open circular cylinder of radius 8 cm so that they have a common axis of symmetry and the rim of the

hemisphere is at one end of the cylinder. The height of the cylinder is k cm. The point O is on the axis

of symmetry and at the end of the cylinder.

OO O

Fig. 4.2 Fig. 4.3

perspective view cross-section cross-section

8 cm 8 cm8 cm 8 cm

k cm 12 cm

end

(i) Show that the centre of mass of the bowl is a distance 64 + k 2

16 + 2k cm from O. [6]

A version of the bowl for the ‘senior dog’ has k = 12 and an end to the cylinder, as shown in Fig. 4.3.

The end is made from the same material as the original bowl.

(ii) Show that the centre of mass of this bowl is a distance 6 13

cm from O. [5]

This bowl is placed on a rough slope inclined at θ to the horizontal.

(iii) Assume that the bowl is prevented from sliding and is on the point of toppling.

Draw a diagram indicating the position of the centre of mass of the bowl with relevant lengths

marked.

Calculate the value of θ . [5]

(iv) If the bowl is not prevented from sliding, determine whether it will slide when placed on the

slope when there is a coefficient of friction between the bowl and the slope of 1.5. [3]

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ADVANCED GCE



OCR Supplied Materials:• 8 page Answer Booklet• Graph paper• MEI Examination Formulae andTables (MF2)

Other Materials Required:None

Monday 11 January 2010

Morning


**44776622**





. Unless otherwise instructed, when a numerical valueis needed, use g = 9.8.





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2

1 (a) An object P, with mass 6 kg and speed 1 m s−1, is sliding on a smooth horizontal table. Object P

explodes into two small parts, Q and R. Q has mass 4 kg and R has mass 2 kg and speed 4 m s−1

in the direction of motion of P before the explosion. This information is shown in Fig. 1.1.

6 kg 4 kg 2 kgP Q R

1 m s–1 4 m s–1

before after

Fig. 1.1

(i) Calculate the velocity of Q. [4]

Just as object R reaches the edge of the table, it collides directly with a small object S of mass3 kg that is travelling horizontally towards R with a speed of 1 m s−1. This information is shown

in Fig. 1.2. The coefficient of restitution in this collision is 0.1.

2 kg 3 kgR S

1 m s–14 m s–1

Fig. 1.2

0.4 m

(ii) Calculate the velocities of R and S immediately after the collision. [6]

The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the

table.

(iii) Calculate the distance apart of R and S when they reach the floor. [3]

(b) A particle of mass m kg bounces off a smooth horizontal plane. The components of velocity of

the particle just before the impact are um s−1 parallel to the plane and vm s−1 perpendicular to

the plane. The coefficient of restitution is e.

Show that the mechanical energy lost in the impact is 12mv2(1 − e2) J. [4]

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4

3 A side view of a free-standing kitchen cupboard on a horizontal floor is shown in Fig. 3.1. The

cupboard consists of: a base CE; vertical ends BC and DE; an overhanging horizontal top AD. The

dimensions, in metres, of the cupboard are shown in the figure. The cupboard and contents have a

weight of 340 N and centre of mass at G.

0.125 2.0

0.90.5

AB

D

EC

G

Fig. 3.1

(i) Calculate the magnitude of the vertical force required at A for the cupboard to be on the point of

tipping in the cases where the force acts

( A) downwards, [3]

( B) upwards. [3]

A force of magnitude QN is now applied at A at an angle of θ to AB, as shown in Fig. 3.2, where

cos θ = 513

(and sin θ = 1213

).

A B D

Fig. 3.2

Q N

(ii) By considering the vertical and horizontal components of the force at A, show that the clockwise

moment of this force about E is 30Q

13 N m. [3]

With the force of magnitude Q N acting as shown in Fig. 3.2, the cupboard is in equilibrium and is on

the point of tipping but not on the point of sliding.

(iii) Show that Q = 221 and that the coefficient of friction between the cupboard base and the floor

must be greater than 58

. [9]

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5

4 In this question, coordinates refer to the axes shown in the figures and the units are centimetres.

Fig. 4.1 shows a lamina KLMNOP shaded. The lamina is made from uniform material and has the

dimensions shown.

x x

y y

O ON N

M ML L

K KP P

Q

40 40

40 40

40 4060 60

80

110

Fig. 4.1 Fig. 4.2

(i) Show that the x -coordinate of the centre of mass of this lamina is 26 and calculate the y-coordinate.

[4]

A uniform thin heavy wire KLMNOPQ is bent into the shape of part of the perimeter of the lamina

KLMNOP with an extension of the side OP to Q, as shown in Fig. 4.2.

(ii) Show that the x -coordinate of the centre of mass of this wire is 23.2 and calculate the y-coordinate.

[5]

The wire is freely suspended from Q and hangs in equilibrium.

(iii) Draw a diagram indicating the position of the centre of mass of the hanging wire and calculate

the angle of QO with the vertical. [4]

A wall-mounted bin with an open top is shown in Fig. 4.3. The centre part has cross-section

KLMNOPQ; the two ends are in the shape of the lamina KLMNOP.

The ends are made from the same uniform, thin material and each has a mass of 1.5 kg. The centre

part is made from different uniform, thin material and has a total mass of 7 kg.

x

y

O N

ML

KP

Q

Fig. 4.3

(iv) Calculate the x - and y-coordinates of the centre of mass of the bin. [5]

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ADVANCED GCE



OCR Supplied Materials:• 8 page Answer Booklet• Graph paper• MEI Examination Formulae and Tables (MF2)

Other Materials Required:• Scientific or graphical calculator

Tuesday 15 June 2010

Morning


**4477

66

22**



• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions.• Do not write in the bar codes.

• You are permitted to use a graphical calculator in this paper.• Final answers should be given to a degree of accuracy appropriate to the context.


. Unless otherwise instructed, when a numerical valueis needed, use g = 9.8.





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2

1 Two sledges P and Q, with their loads, have masses of 200 kg and 250 kg respectively and are sliding

on a horizontal surface against negligible resistance. There is an inextensible light rope connecting

the sledges that is horizontal and parallel to the direction of motion.

Fig. 1 shows the initial situation with both sledges travelling with a velocity of 5i m s−1.

a a a a a a a a a a a a a a a a a a a a a a a a a a a a a











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arope

sledge P200 kg

sledge Q250 kg

i

5 m si –1 5 m si –1

Fig. 1

A mechanism on Q jerks the rope so that there is an impulse of 250i N s on P.

(i) Show that the new velocity of P is 6.25i m s−1 and find the new velocity of Q. [4]

There is now a direct collision between the sledges and after the impact P has velocity 4.5 i m s−1.

(ii) Show that the velocity of Q becomes 5.4i m s−1.

Calculate the coefficient of restitution in the collision. [6]

Before the rope becomes taut again, the velocity of P is increased so that it catches up with Q. Thisis done by throwing part of the load from sledge P in the −i direction so that P’s velocity increases

to 5.5i m s−1. The part of the load thrown out is an object of mass 20 kg.

(iii) Calculate the speed of separation of the object from P. [5]

When the sledges directly collide again they are held together and move as a single object.

(iv) Calculate the common velocity of the pair of sledges, giving your answer correct to 3 significant

figures. [2]

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3

2

15

1540

40

y

x

O

P

Q

R

S

A

B

50

200

not to

scale

the lengths arein centimetres

Fig. 2

Fig. 2 shows a stand on a horizontal floor and horizontal and vertical coordinate axes O x and O y. The

stand is modelled as

• a thin uniform rectangular base PQRS, 30 cm by 40 cm with mass 15 kg; the sides QR and PSare parallel to O x ,

• a thin uniform vertical rod of length 200 cm and mass 3 kg that is fixed to the base at O, themid-point of PQ and the origin of coordinates,

• a thin uniform top rod AB of length 50 cm and mass 2 kg; AB is parallel to O x .

Coordinates are referred to the axes shown in the figure.

(i) Calculate the coordinates of the centre of mass of the stand. [5]

A small object of mass 5 kg is fixed to the rod AB at a distance of 40 cm from A.

(ii) Show that the coordinates of the centre of mass of the stand with the object are (22, 68). [2]

The stand is tilted about the edge PQ until it is on the point of toppling. The angle through which the

stand is tilted is called ‘the angle of tilt’. This procedure is repeated about the edges QR and RS.

(iii) Making your method clear, determine which edge requires the smallest angle of tilt for the stand

to topple. [4]

The small object is removed. A light string is attached to the stand at A and pulled at an angle of 50◦

to the downward vertical in the plane O xy in an attempt to tip the stand about the edge RS.

(iv) Assuming that the stand does not slide, find the tension in the string when the stand is about to

turn about the edge RS. [7]

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4

3 Fig. 3 shows a framework in a vertical plane constructed of light, rigid rods AB, BC, CD, DA and

BD. The rods are freely pin-jointed to each other at A, B, C and D and to a vertical wall at A. ABCD

is a parallelogram with AD horizontal and BD vertical; the dimensions of the framework, in metres,

are shown. There is a vertical load of 300 N acting at C and a vertical wire attached to D, with tension

T N, holds the framework in equilibrium. The horizontal and vertical forces, X N and Y N, acting on

the framework at A due to the wall are also shown.

2

1

1

2

A

B C

D

300 N

T N

X N

Y N

Fig. 3

(i) Show that T = 600 and calculate the values of X and Y . [4]

(ii) Draw a diagram showing all the forces acting on the framework, and also the internal forces in

the rods. [2]

(iii) Calculate the internal forces in the five rods, indicating whether each rod is in tension or

compression (thrust). (You may leave answers in surd form. Your working in this part should

correspond to your diagram in part (ii).) [9]

Suppose that the vertical wire is attached at B instead of D and that the framework is still in equilibrium.

(iv) Without doing any further calculations, state which four of the rods have the same internal forces

as in part (iii) and say briefly why this is the case. Determine the new force in the fifth rod. [4]

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5

4 A box of mass 16 kg is on a uniformly rough horizontal floor with an applied force of fixed direction

but varying magnitude P N acting as shown in Fig. 4. You may assume that the box does not tip for

any value of P. The coefficient of friction between the box and the floor is µ .

16 kg

P N35°

Fig. 4

Initially the box is at rest and on the point of slipping with P = 58.

(i) Show that µ is about 0.25. [5]

In the rest of this question take µ to be exactly 0.25.

The applied force on the box is suddenly increased so that P = 70 and the box moves against friction

with the floor and another horizontal retarding force, S . The box reaches a speed of 1.5 m s−1 from

rest after 5 seconds; during this time the box slides 3 m.

(ii) Calculate the work done by the applied force of 70 N and also the average power developed by

this force over the 5 seconds. [4]

(iii) By considering the values of time, distance and speed, show that an object starting from restthat travels 3 m while reaching a speed of 1.5 m s−1 in 5 seconds cannot be moving with constant

acceleration. [2]

The reason that the acceleration is not constant is that the retarding force S is not constant.

(iv) Calculate the total work done by the retarding force S . [7]

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ADVANCED GCE


QUESTION PAPER

Candidates answer on the printed answer book.

OCR supplied materials:• Printed answer book 4762• MEI Examination Formulae andTables (MF2)

Other materials required:• Scientific or graphical calculator

Monday 10 January 2011

Morning


INSTRUCTIONS TO CANDIDATESThese instructions are the same on the printed answer book and the question paper.

• The question paper will be found in the centre of the printed answer book.• Write your name, centre number and candidate number in the spaces provided on the printed answer book.

Please write clearly and in capital letters.• Write your answer to each question in the space provided in the printed answer book. Additional paper

may be used if necessary but you must clearly show your candidate number, centre number and questionnumber(s).

• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully. Make sure you know what you have to do before starting your answer.• Answer all the questions.• Do not write in the bar codes.• You are permitted to use a scientific or graphical calculator in this paper.• Final answers should be given to a degree of accuracy appropriate to the context.

• The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when a numerical value isneeded, use g = 9.8.

INFORMATION FOR CANDIDATESThis information is the same on the printed answer book and the question paper.

• The number of marks is given in brackets [ ] at the end of each question or part question on the question paper.• You are advised that an answer may receive no marks unless you show sufficient detail of the working to

indicate that a correct method is being used.• The total number of marks for this paper is 72.• The printed answer book consists of 12 pages. The question paper consists of 8 pages. Any blank pages are

indicated.

INSTRUCTION TO EXAMS OFFICER / INVIGILATOR

• Do not send this question paper for marking; it should be retained in the centre or destroyed.


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2

1 Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined

at an angle α to the horizontal, where cos α = 0.8. The coefficient of friction between A and the slope

is 0.85.

A A

B

2.5 kg 2.5 kg

1.5 kg

Fig. 1.1 Fig. 1.2

(i) Calculate the maximum possible frictional force between A and the slope.

Show that A will remain at rest. [6]

With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has

a speed of 16 m s−1 when it collides with A. In this collision the coefficient of restitution is 0.4, the

impulses are parallel to the slope and linear momentum parallel to the slope is conserved.

(ii) Show that the velocity of A immediately after the collision is 8.4 m s−1 down the slope.

Find the velocity of B immediately after the collision. [6]

(iii) Calculate the impulse on B in the collision. [3]

The blocks do not collide again.

(iv) For what length of time after the collision does A slide before it comes to rest? [4]

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3

2 (a) A firework is instantaneously at rest in the air when it explodes into two parts. One part is the

body B of mass 0.06 kg and the other a cap C of mass 0.004 kg. The total kinetic energy given

to B and C is 0.8 J. B moves off horizontally in the i direction.

By considering both kinetic energy and linear momentum, calculate the velocities of B and C

immediately after the explosion. [8]

(b) A car of mass 800 kg is travelling up some hills.

In one situation the car climbs a vertical height of 20 m while its speed decreases from 30 m s−1

to 12 m s−1. The car is subject to a resistance to its motion but there is no driving force and the

brakes are not being applied.

(i) Using an energy method, calculate the work done by the car against the resistance to its

motion. [4]

In another situation the car is travelling at a constant speed of 18 m s−1 and climbs a vertical

height of 20 m in 25 s up a uniform slope. The resistance to its motion is now 750 N.

(ii) Calculate the power of the driving force required. [5]

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4

3

A D

CB

50 N

45 N

X N

Y N

R N

Fig. 3

Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal,

light, rigid rods AB, AD, BC, BD and CD so that ABD and BCD are equilateral triangles of side 2 m.AD and BC are horizontal.

The rods are freely pin-jointed to each other at A, B, C and D. The pin-joint at A is fixed to a wall

and the pin-joint at B rests on a smooth horizontal support.

Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and

a horizontal force of 50 N applied at D; the normal reaction of the support on the framework at B is

RN; horizontal and vertical forces X N and Y N act at A.

(i) Write down equations for the horizontal and vertical equilibrium of the framework. [2]

(ii) Show that R = 135 and Y = 90. [3]

(iii) On the diagram in your printed answer book, show the forces internal to the rods acting on the

pin-joints. [2]

(iv) Calculate the forces internal to the five rods, stating whether each rod is in tension or compression

(thrust). [You may leave your answers in surd form. Your working in this part should correspond

to your diagram in part (iii).] [10]

(v) Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system

remains in equilibrium in the same position.

By considering the equilibrium at C, show that the forces in rods CD and BC are not changed.

[2]

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5

4 You are given that the centre of mass, G, of a uniform lamina in

the shape of an isosceles triangle lies on its axis of symmetry in

the position shown in Fig. 4.1.

Fig. 4.2 shows the cross-section OABCD of a prism made from

uniform material. OAB is an isosceles triangle, where OA = AB,

and OBCD is a rectangle. The distance OD is h cm, where h can

take various positive values. All coordinates refer to the axes O x

and O y shown. The units of the axes are centimetres.

P

G

Q

PG = 2 GQ

Fig. 4.1

x

y

O

D

C

A

B

2.5

5

6–h

Fig. 4.2

(i) Write down the coordinates of the centre of mass of the triangle OAB. [1]

(ii) Show that the centre of mass of the region OABCD is 12 − h2

2(h + 3) , 2.5. [6]The x -axis is horizontal.

The prism is placed on a horizontal plane in the position shown in Fig. 4.2.

(iii) Find the values of h for which the prism would topple. [3]

The following questions refer to the case where h = 3 with the prism held in the position shown in

Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism

is 15 N. You should assume that the prism does not slide.

(iv) Suppose that the prism is held in this position by a vertical force applied at A. Given that the

prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]

(v) Suppose instead that the prism is held in this position by a force in the plane of the cross-section

OABCD, applied at 30◦ below the horizontal at C, as shown in Fig. 4.3. Given that the prism is

on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]

C

30°

Fig. 4.3

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8

Copyright Information

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders

whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright

Acknowledgements Booklet. This is produced for each series of examinations a

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